Problem: The sum of the $x$-coordinates of the vertices of a triangle in the Cartesian plane equals $10$.  Find the sum of the $x$-coordinates of the midpoints of the sides of the triangle.
Explanation: Let the $x$-coordinates of the vertices be $a,b,c$.  Then the $x$-coordinates of the midpoints of the sides are $\frac{a+b}2,\frac{a+c}2,\frac{b+c}2$.  The sum of these equals $\frac{2a+2b+2c}2=a+b+c$.  Thus the desired answer is $\boxed{10}$.